Question

in which quadrant would \\( \theta \\) be if \\( \tan \theta=-\sqrt{3} \\) and \\( \sin \theta>0 \\)? (1 point) quadrant iv quadrant i quadrant ii quadrant iii

Explanation:

Step1: Recall the sign of trigonometric functions in each quadrant

• In Quadrant I: \(\sin\theta> 0\), \(\cos\theta>0\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}>0\)
• In Quadrant II: \(\sin\theta> 0\), \(\cos\theta<0\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}<0\)
• In Quadrant III: \(\sin\theta< 0\), \(\cos\theta<0\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}>0\)
• In Quadrant IV: \(\sin\theta< 0\), \(\cos\theta>0\), \(\tan\theta=\frac{\sin\theta}{\cos\theta}<0\)

Step2: Analyze the given conditions

We are given that \(\tan\theta =-\sqrt{3}<0\) and \(\sin\theta>0\).
From the sign - rules of trigonometric functions:
If \(\tan\theta=\frac{\sin\theta}{\cos\theta}<0\), then \(\sin\theta\) and \(\cos\theta\) have opposite signs.
Since \(\sin\theta>0\), then \(\cos\theta<0\)

Answer:

Quadrant II